Boil the chayote until they soften (approximately one hour). Crumble the cuajada and add the tomato, pepper, onion, and a bouillon cube for taste. Once the chayotes have cooled, cut them in half and remove the excess from the centers. Mash the extra chayote and add the cuajada mix that has already been prepared. Stuff the chayote shells with the mixture and cover the shells with the beaten egg. Place in the oven at 400 degrees for 15 to 20 minutes or until the outside of the shells have a golden color.